GCSE Maths Practice: factors-and-multiples

Question 4 of 10

This GCSE Maths question focuses on finding the Highest Common Factor (HCF) of two numbers, an essential concept for simplifying fractions and solving ratio problems.

‹ ›

\( \begin{array}{l}\text{What is the highest common factor (HCF) of 9 and 12?}\end{array} \)

Choose one option:

To find the HCF, list the factors of each number or use prime factorisation. The largest number that divides both exactly is the answer.

Understanding Common Factors and Multiples

In GCSE Maths, understanding the difference between factors and multiples is essential. Factors are the numbers that divide exactly into another number, while multiples are the results you get when you multiply a number by 1, 2, 3, and so on. This question checks whether you can reason about how two numbers are related through their shared factors.

Step-by-Step Method to Find the HCF

  1. Write down all the factors of each number.
  2. Identify which numbers appear in both lists — these are the common factors.
  3. The largest of these is the Highest Common Factor (HCF).

Worked Examples (Different Values)

  • Example 1: Find the HCF of 9 and 12.
    9 → 1, 3, 9.
    12 → 1, 2, 3, 4, 6, 12.
    Common: 1, 3 → HCF = 3.
  • Example 2: Find the HCF of 15 and 20.
    15 → 1, 3, 5, 15.
    20 → 1, 2, 4, 5, 10, 20.
    Common: 1, 5 → HCF = 5.
  • Example 3: Find the HCF of 18 and 27.
    18 → 1, 2, 3, 6, 9, 18.
    27 → 1, 3, 9, 27.
    Common: 1, 3, 9 → HCF = 9.

Prime Factorisation Method

Another reliable way to find common factors is through prime factorisation. This means breaking numbers into their prime components.

Example: 9 = 3 × 3, 12 = 2 × 2 × 3. The shared prime is 3 → HCF = 3.

Using Divisibility Rules

Divisibility rules make it quicker to identify common factors:

  • Divisible by 2: Even numbers end in 0, 2, 4, 6, or 8.
  • Divisible by 3: The sum of the digits is divisible by 3.
  • Divisible by 5: Numbers end in 0 or 5.

For example, 9 and 12 both have digit sums (9 and 3) divisible by 3, confirming they share 3 as a factor.

Common Mistakes

  • Confusing factors with multiples: Factors are smaller numbers that divide; multiples are larger numbers created by multiplying.
  • Not listing all factors: Skipping one can lead to missing the HCF.
  • Mixing HCF and LCM: HCF is the greatest number that divides both; LCM is the smallest number both divide into.

Real-Life Applications

Finding common factors helps in simplifying problems involving sharing or grouping. For instance:

  • Dividing Items Evenly: If 9 red and 12 blue balloons must be arranged into identical sets, each set can have 3 of each colour, based on the HCF = 3.
  • Fractions: Simplifying fractions like 9⁄12 → divide numerator and denominator by 3 → 3⁄4.
  • Ratios: Ratios such as 9:12 simplify to 3:4 by dividing both sides by their HCF.

Frequently Asked Questions

Q1: Can the HCF ever be one of the numbers?
A: Yes, if one number divides exactly into the other. For example, HCF(5, 10) = 5.

Q2: What if two numbers have no common factors other than 1?
A: They are called co-prime or relatively prime numbers.

Q3: How can I check my answer quickly?
A: Divide both numbers by your HCF — you should get whole numbers with no remainder.

GCSE Study Tip

When solving factor or multiple questions, always identify which type the question is asking for. Remember: factors are smaller or equal, multiples are equal or larger. Building this habit prevents confusion in mixed practice sets.

Summary

To find the Highest Common Factor (HCF) of two numbers, list or calculate their factors and find the largest shared one. For 9 and 12, the HCF is 3. This principle supports key GCSE topics such as simplifying fractions, comparing ratios, and solving common multiple problems.